Syntactic Methods

from Mathematical Girl: Gödel’s Incompleteness Theorem

• Syntactic Methods
  • Antonym: Semantic Methods

    • The difference is whether to use truth values.

      • I’m not quite sure.
      • In syntactic methods, there is no concept of what is true or false, it’s just about syntax.
      • Even if we define something like “axioms,” they are neither true nor false, they are just “axioms.”
  • Understanding it as processing sequences of symbols as sequences of symbols mechanically to do mathematics, like programming (for now).
    • This gives me a familiar impression because it’s like programming.
  • It’s about “doing mathematics in mathematics.”
  • Dealing with formal systems.
  • Creating a miniature model of mathematics called “formal system H.”
    • Wow~

    • Defining the symbol IMPLY: $(x)\to (y)$ as $\neg (x)\vee (y)$

      • It’s like an alias or wrapper in programming.
      • It can be replaced as a string without being interpreted, similar to C++‘s using.
      • However, I’m not quite sure why this definition is made.
        • Maybe it’s not necessary to think about it when dealing with formal systems, but I want to understand the underlying intention if I want to comprehend it.
        • Semantically, if we consider “if a then b,” its truth value matches the truth value of “not a or b.”
          • But I still don’t get it.
          • Thinking about it in terms of “If apples are delicious, then apple pie is also delicious,”
            • Apples are delicious & apple pie is delicious: true
            • Apples are delicious & apple pie is not delicious: false
            • Apples are not delicious & apple pie is delicious: true
            • Apples are not delicious & apple pie is not delicious: true
          • Does it mean that anything is okay as long as a is false?
            • I can kind of understand it.
    • In formal systems, by mechanically executing operations on defined strings, something like “inference” in semantics can be done.
      • It is defined as a formal modus ponens